Question: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{n^3 + n^2 - 12n}{n^3 + 12n^2 + 32n}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {n(n^2 + n - 12)} {n(n^2 + 12n + 32)} $ $ z = \dfrac{n}{n} \cdot \dfrac{n^2 + n - 12}{n^2 + 12n + 32} $ Simplify: $ z = \dfrac{n^2 + n - 12}{n^2 + 12n + 32}$ Since we are dividing by $n$ , we must remember that $n \neq 0$ Next factor the numerator and denominator. $ z = \dfrac{(n + 4)(n - 3)}{(n + 4)(n + 8)}$ Assuming $n \neq -4$ , we can cancel the $n + 4$ $ z = \dfrac{n - 3}{n + 8}$ Therefore: $ z = \dfrac{ n - 3 }{ n + 8 }$, $n \neq -4$, $n \neq 0$